We characterize equicontinuous Delone dynamical systems as those
coming from Delone sets with strongly almost periodic Dirac combs.
Within the class of systems with finite local complexity, the only
equicontinuous systems are then shown to be the crystallographic
ones. On the other hand, within the class without finite local
complexity, we exhibit examples of equicontinuous minimal Delone
dynamical systems that are not crystallographic.
Our results solve the problem posed by Lagarias as to whether a Delone
set whose Dirac comb is strongly almost periodic must be
crystallographic.